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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldmcoss | Structured version Visualization version GIF version |
Description: Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.) |
Ref | Expression |
---|---|
eldmcoss | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmcoss3 35725 | . . 3 ⊢ dom ≀ 𝑅 = dom ◡𝑅 | |
2 | 1 | eleq2i 2904 | . 2 ⊢ (𝐴 ∈ dom ≀ 𝑅 ↔ 𝐴 ∈ dom ◡𝑅) |
3 | eldmcnv 35634 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) | |
4 | 2, 3 | syl5bb 285 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∃wex 1780 ∈ wcel 2114 class class class wbr 5052 ◡ccnv 5540 dom cdm 5541 ≀ ccoss 35485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pr 5316 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-br 5053 df-opab 5115 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-coss 35691 |
This theorem is referenced by: eldmcoss2 35731 eldm1cossres 35732 |
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